In Exercises 43–54, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 1+3+5+⋯+ (2n−1)

Verified step by step guidance

1. Recognize the pattern of the sum: the terms are 1, 3, 5, ..., which are consecutive odd numbers. Each term can be expressed as \$2i - 1\(, where \)i$ is the index of summation starting from 1.
2. Identify the number of terms in the sum. Since the last term is given as \$2n - 1\(, the number of terms is \)n$.
3. Write the summation notation using the index \(i\) starting at 1 and going up to \(n\), summing the expression for each term.
4. Express the sum as \(\sum_{i=1}^{n} (2i - 1)\), which compactly represents the sum of the first \(n\) odd numbers.
5. This summation notation captures the entire sum \$1 + 3 + 5 + \cdots + (2n - 1)$ in a concise mathematical form.

Key Concepts

Summation Notation

Summation notation is a concise way to represent the sum of a sequence of terms using the sigma symbol (∑). It includes an index of summation, lower and upper limits, and a general term formula. For example, ∑_{i=1}^n a_i represents the sum of terms a_i from i=1 to i=n.

Arithmetic Sequences

An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. In this problem, the sequence 1, 3, 5, ..., (2n−1) is arithmetic with a common difference of 2. Understanding this helps identify the general term formula for the sequence.

General Term of the Sequence

The general term formula expresses the nth term of a sequence as a function of n or the index i. For the sequence of odd numbers 1, 3, 5, ..., the nth term is given by 2i−1. This formula is essential to write the sum in summation notation.